General Motion of Long Waves. 351 



It must be admitted that it does not appear possible to pro- 

 ceed further with the general discussion of this equation ; and 

 that might be expected from the very extensive range of 

 motions to which the equation applies. It will still be possible 

 to learn something from the equation in special questions. 



§ 2. But first it seems desirable to test the correctness by 

 considering the simpler case of progressive motion. 



For this purpose write F(#, i) in the form ¥(ct + x) } so 



that F = cF'. 



The equation (5) then reduces to 



(c 2 - •^U 2 F lv = 3F /2 F // -6cF / F / ' + 2{c 9 -ffh)F", . (6) 



which agrees with that, viz. 



if- f ) Nf M =f*-W*. + Hc*- 9 h)f, 



obtained in the paper referred to above, when it is remem- 

 bered that we have now performed a differentiation in excess. 

 It is easily seen that, if we neglect the term 3F /2 F", as we 

 may generally do, one form of solution is 



F (ct +.!•)= a. tank nti(ct + .v), .... (7) 

 where 



c* — qh 

 ma= *~. 



c 



c —gh 



»m-. 



(>-$ 



ghS 



a form which suits the circumstances of a low Solitary Wave. 



§ 3, I shall now select a special question on which to employ 

 the general equation (5). 



In his experiments on the Solitary Wave, Scott Russell 

 found that, with moderate elevations, it was directly reflected 

 from a vertical wall so as to travel without appreciable change 

 of profile or of rate of progression, and, in his experimental 

 investigations to secure exact measurements, the wave was 

 always so reflected at the ends of the trough along which 

 it ran. 



For the purpose of investigating the mathematical con- 

 ditions of such a reflexion, the ordinary treatment of progressive 

 motion will not suffice. 



To represent the circumstances, let us replace in (5) 

 F(x,t) by 



F^ + ^O + F^-.*), ...... (8) 



